《孫子算經卷上》 "Sun Tzŭ's Computational Classic: Volume I"
§6. Densities of metals etc.

This section gives the densities of some precious & base metals, along with those of jade and an unspecific "stone/rock".

Translation

Chinese source text: Version A, Version B, Version C, Version D.
Unless noted otherwise, I follow the text from Version D, 《知不足齋叢書》本.

Source text Target text Notes
黃金方寸重一斤、
白金方寸重一十四兩、
玉方寸重一十二兩、		 [A] cubic inch of gold weigheth one catty.
[A] cubic inch of silver weigheth fourteen taels.
[A] cubic inch of jade weigheth twelve taels.
  • 黃金： gold; lit. yellow metal
  • Version B has 白銀 for 白金.
  • 白金： silver; lit. white metal
  • For jade, Version B has 一十兩 for 一十二兩.
銅方寸重七兩半、
鉛方寸重九兩半、
鐵方寸重六兩、
石方寸重三兩。		 [A] cubic inch of copper weigheth seven taels [and an] half.
[A] cubic inch of lead weigheth nine taels [and an] half.
[A] cubic inch of iron weigheth six taels.
[A] cubic inch of stone weigheth three taels.
  • 銅： copper
    Or, perhaps brass.
  • For iron, Version B has 七兩 for 六兩.
  • 石： stone; or rock

Extended commentary

Let us do a quick comparison of the densities. For the modern density of 鐵 (iron) I have used cast iron, since that value is available, and the Chinese probably couldn't get pure iron back then. 石 (stone) I have excluded from the analysis since it does not adequately pinpoint a specific material.

Material Sun Tzŭ value
\rho_\text{s} / (\unit{tael} \unit{inch}^{-3})		 Modern value
\rho_\text{m} / (\unit{g} \unit{cm}^{-3})		 \dfrac{ \rho_\text{s} / (\unit{tael} \unit{inch}^{-3}) }{ \rho_\text{m} / (\unit{g} \unit{cm}^{-3}) } Reference
金 Gold 16 19.29 0.83 Engineering ToolBox
銀 Silver 14 10.5 1.33 Engineering ToolBox
玉 Jade 12  3 (roughly) 4 (roughly) Wikipedia Wikipedia
銅 Copper  7.5  8.79 0.85 Engineering ToolBox
鉛 Lead  9.5 11.35 0.84 Engineering ToolBox
鐵 Cast iron  6  7.2 0.83 Engineering ToolBox

Apart from silver and jade, the densities appear to fit quite well, so Sun Tzŭ either fluked the measurements for gold, copper, lead, and iron or he was actually a decent experimentalist but buggered up the measurements for silver and jade. Or maybe he only had access to very impure silver and jade:

Scatter plot of Sun Tzŭ versus modern densities for gold, silver, jade, copper, lead, and cast iron.

Discarding the outliers 銀 (silver) and 玉 (jade) and forcing a least squares fit through the origin, we get

\frac{\rho_\text{s}}{\unit{tael} \unit{inch}^{-3}} = 0.834229 \cdot \frac{\rho_\text{m}}{\unit{g} \unit{cm}^{-3}}

with R2 = 0.999915; the slope has standard error 0.004453.

Assuming that Sun Tzŭ's \rho_\text{s} and toady's \rho_\text{m} are referring to the same material, we should have that \rho_\text{s} = \rho_\text{m}. Therefore we obtain the following experimental conversion relation between Sun Tzŭ and modern density units:

1 \unit{tael} \unit{inch}^{-3} = (1.1987 \pm 0.0064) \unit{g} \unit{cm}^{-3}.

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume I §6". <https://yawnoc.github.io/sun-tzu/i/6> Accessed yyyy-mm-dd.